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Green's function method is a common method in mathematical and physical equations.

Physically speaking, a mathematical and physical equation represents the relationship between a specific "field" and its "source". For example, heat conduction equation represents the relationship between temperature field and heat source, Poisson equation represents the relationship between electrostatic field and charge distribution and so on. In this way, when the source is decomposed into the superposition of many point sources, if we can try to know the field generated by the point sources, we can use the superposition principle to find the field of any source under the same boundary conditions.

Grimm Han Shu

Green's function

Green's function

Green's function, also known as source function or influence function, is an important function in physics in mathematical physics methods. It was put forward by Englishman G Green in 1828.

The definition of Green's function used in the single quantum theory in physics is slightly extended. It satisfies the equation: (-) (,) = (-), where is the single-particle Hamiltonian, which can include external field and impurity potential. Single Green's function has important applications in the study of disordered systems, such as using average matrix approximation and coherent potential approximation to find the density of states.

Since 1960s, Green's function of many-body quantum theory has become a powerful tool to study condensed matter theory. At present, Green's function in physics often refers to the multi-body Green's function used to study the system composed of a large number of interacting particles. Many-body Green's function represents the probability amplitude of a particle joining the system at a certain time and place and then appearing at the same time. Green's function describes the propagation behavior of particles, also known as propagator.

In order to study the equilibrium behavior of multi-particle system when it is greater than absolute zero, the temperature Green's function is introduced. Because there is a formal correspondence between the reciprocal of temperature and virtual time, the temperature Green's function is also called virtual time Green's function. In order to study the non-equilibrium behavior of 0K, [kg2] introduced the time Green's function and closed-circuit Green's function of 0K.

In quantum field theory, Green's function often appears when calculating the matrix elements of specific physical processes, and its physical meaning also represents the probability amplitude of particle propagation. Because many-body Green's function corresponds to it at 0 k, Feynman diagram method in quantum field theory (see Feynman diagram) can also be used for many-body Green's function. Renormalization group method has also been applied in the study of condensed matter in recent ten years, such as Kondo effect and one-dimensional conductor.